LMFDB - Embedded newform 6039.2.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} + \cdots - 35 $ x^13 - 4x^12 - 11x^11 + 57x^10 + 28x^9 - 290x^8 + 51x^7 + 644x^6 - 259x^5 - 640x^4 + 274x^3 + 256x^2 - 74x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.17594$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17594 q^{2} -0.617176 q^{4} -3.63888 q^{5} -0.818481 q^{7} -3.07763 q^{8} +O(q^{10})$	\(q+1.17594 q^{2} -0.617176 q^{4} -3.63888 q^{5} -0.818481 q^{7} -3.07763 q^{8} -4.27908 q^{10} -1.00000 q^{11} +3.69107 q^{13} -0.962480 q^{14} -2.38474 q^{16} -2.16275 q^{17} +6.52386 q^{19} +2.24583 q^{20} -1.17594 q^{22} +2.86705 q^{23} +8.24142 q^{25} +4.34046 q^{26} +0.505147 q^{28} +8.99583 q^{29} +4.22270 q^{31} +3.35096 q^{32} -2.54326 q^{34} +2.97835 q^{35} -9.05727 q^{37} +7.67163 q^{38} +11.1991 q^{40} -5.62807 q^{41} +4.76569 q^{43} +0.617176 q^{44} +3.37146 q^{46} -6.54699 q^{47} -6.33009 q^{49} +9.69137 q^{50} -2.27804 q^{52} -2.05593 q^{53} +3.63888 q^{55} +2.51898 q^{56} +10.5785 q^{58} -3.99937 q^{59} +1.00000 q^{61} +4.96562 q^{62} +8.70999 q^{64} -13.4313 q^{65} -8.12113 q^{67} +1.33480 q^{68} +3.50235 q^{70} +6.74619 q^{71} +4.15354 q^{73} -10.6508 q^{74} -4.02637 q^{76} +0.818481 q^{77} -15.8018 q^{79} +8.67778 q^{80} -6.61825 q^{82} -1.52507 q^{83} +7.87000 q^{85} +5.60414 q^{86} +3.07763 q^{88} +2.32189 q^{89} -3.02107 q^{91} -1.76947 q^{92} -7.69883 q^{94} -23.7395 q^{95} +3.41810 q^{97} -7.44378 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.17594	0.831512	0.415756	−	0.909476i	$-0.363517\pi$
$2$	1.17594	0.831512	0.415756	+	0.909476i	$0.363517\pi$
$3$	0	0
$3$	0	0
$4$	−0.617176	−0.308588
$5$	−3.63888	−1.62735	−0.813677	−	0.581317i	$-0.802538\pi$
$5$	−3.63888	−1.62735	−0.813677	+	0.581317i	$0.802538\pi$
$6$	0	0
$7$	−0.818481	−0.309357	−0.154678	−	0.987965i	$-0.549434\pi$
$7$	−0.818481	−0.309357	−0.154678	+	0.987965i	$0.549434\pi$
$8$	−3.07763	−1.08811
$9$	0	0
$10$	−4.27908	−1.35316
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.69107	1.02372	0.511859	−	0.859069i	$-0.328957\pi$
$13$	3.69107	1.02372	0.511859	+	0.859069i	$0.328957\pi$
$14$	−0.962480	−0.257234
$15$	0	0
$16$	−2.38474	−0.596185
$17$	−2.16275	−0.524545	−0.262273	−	0.964994i	$-0.584472\pi$
$17$	−2.16275	−0.524545	−0.262273	+	0.964994i	$0.584472\pi$
$18$	0	0
$19$	6.52386	1.49668	0.748338	−	0.663318i	$-0.230852\pi$
$19$	6.52386	1.49668	0.748338	+	0.663318i	$0.230852\pi$
$20$	2.24583	0.502182
$21$	0	0
$22$	−1.17594	−0.250710
$23$	2.86705	0.597820	0.298910	−	0.954281i	$-0.403377\pi$
$23$	2.86705	0.597820	0.298910	+	0.954281i	$0.403377\pi$
$24$	0	0
$25$	8.24142	1.64828
$26$	4.34046	0.851234
$27$	0	0
$28$	0.505147	0.0954638
$29$	8.99583	1.67048	0.835242	−	0.549883i	$-0.185328\pi$
$29$	8.99583	1.67048	0.835242	+	0.549883i	$0.185328\pi$
$30$	0	0
$31$	4.22270	0.758419	0.379209	−	0.925311i	$-0.376196\pi$
$31$	4.22270	0.758419	0.379209	+	0.925311i	$0.376196\pi$
$32$	3.35096	0.592371
$33$	0	0
$34$	−2.54326	−0.436165
$35$	2.97835	0.503433
$36$	0	0
$37$	−9.05727	−1.48901	−0.744503	−	0.667619i	$-0.767313\pi$
$37$	−9.05727	−1.48901	−0.744503	+	0.667619i	$0.767313\pi$
$38$	7.67163	1.24450
$39$	0	0
$40$	11.1991	1.77074
$41$	−5.62807	−0.878957	−0.439478	−	0.898253i	$-0.644837\pi$
$41$	−5.62807	−0.878957	−0.439478	+	0.898253i	$0.644837\pi$
$42$	0	0
$43$	4.76569	0.726761	0.363380	−	0.931641i	$-0.381623\pi$
$43$	4.76569	0.726761	0.363380	+	0.931641i	$0.381623\pi$
$44$	0.617176	0.0930428
$45$	0	0
$46$	3.37146	0.497095
$47$	−6.54699	−0.954976	−0.477488	−	0.878638i	$-0.658453\pi$
$47$	−6.54699	−0.954976	−0.477488	+	0.878638i	$0.658453\pi$
$48$	0	0
$49$	−6.33009	−0.904298
$50$	9.69137	1.37057
$51$	0	0
$52$	−2.27804	−0.315907
$53$	−2.05593	−0.282404	−0.141202	−	0.989981i	$-0.545097\pi$
$53$	−2.05593	−0.282404	−0.141202	+	0.989981i	$0.545097\pi$
$54$	0	0
$55$	3.63888	0.490666
$56$	2.51898	0.336613
$57$	0	0
$58$	10.5785	1.38903
$59$	−3.99937	−0.520673	−0.260336	−	0.965518i	$-0.583833\pi$
$59$	−3.99937	−0.520673	−0.260336	+	0.965518i	$0.583833\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	4.96562	0.630634
$63$	0	0
$64$	8.70999	1.08875
$65$	−13.4313	−1.66595
$66$	0	0
$67$	−8.12113	−0.992154	−0.496077	−	0.868278i	$-0.665227\pi$
$67$	−8.12113	−0.992154	−0.496077	+	0.868278i	$0.665227\pi$
$68$	1.33480	0.161868
$69$	0	0
$70$	3.50235	0.418610
$71$	6.74619	0.800625	0.400313	−	0.916379i	$-0.368902\pi$
$71$	6.74619	0.800625	0.400313	+	0.916379i	$0.368902\pi$
$72$	0	0
$73$	4.15354	0.486135	0.243068	−	0.970009i	$-0.421846\pi$
$73$	4.15354	0.486135	0.243068	+	0.970009i	$0.421846\pi$
$74$	−10.6508	−1.23813
$75$	0	0
$76$	−4.02637	−0.461856
$77$	0.818481	0.0932745
$78$	0	0
$79$	−15.8018	−1.77784	−0.888918	−	0.458065i	$-0.848543\pi$
$79$	−15.8018	−1.77784	−0.888918	+	0.458065i	$0.848543\pi$
$80$	8.67778	0.970205
$81$	0	0
$82$	−6.61825	−0.730863
$83$	−1.52507	−0.167398	−0.0836990	−	0.996491i	$-0.526673\pi$
$83$	−1.52507	−0.167398	−0.0836990	+	0.996491i	$0.526673\pi$
$84$	0	0
$85$	7.87000	0.853621
$86$	5.60414	0.604310
$87$	0	0
$88$	3.07763	0.328076
$89$	2.32189	0.246119	0.123060	−	0.992399i	$-0.460729\pi$
$89$	2.32189	0.246119	0.123060	+	0.992399i	$0.460729\pi$
$90$	0	0
$91$	−3.02107	−0.316694
$92$	−1.76947	−0.184480
$93$	0	0
$94$	−7.69883	−0.794074
$95$	−23.7395	−2.43562
$96$	0	0
$97$	3.41810	0.347055	0.173528	−	0.984829i	$-0.444483\pi$
$97$	3.41810	0.347055	0.173528	+	0.984829i	$0.444483\pi$
$98$	−7.44378	−0.751935
$99$	0	0
$100$	−5.08641	−0.508641
$101$	4.66357	0.464042	0.232021	−	0.972711i	$-0.425466\pi$
$101$	4.66357	0.464042	0.232021	+	0.972711i	$0.425466\pi$
$102$	0	0
$103$	−1.34770	−0.132793	−0.0663964	−	0.997793i	$-0.521150\pi$
$103$	−1.34770	−0.132793	−0.0663964	+	0.997793i	$0.521150\pi$
$104$	−11.3597	−1.11391
$105$	0	0
$106$	−2.41764	−0.234822
$107$	−10.6764	−1.03213	−0.516064	−	0.856550i	$-0.672603\pi$
$107$	−10.6764	−1.03213	−0.516064	+	0.856550i	$0.672603\pi$
$108$	0	0
$109$	−12.7736	−1.22349	−0.611744	−	0.791056i	$-0.709532\pi$
$109$	−12.7736	−1.22349	−0.611744	+	0.791056i	$0.709532\pi$
$110$	4.27908	0.407995
$111$	0	0
$112$	1.95186	0.184434
$113$	6.42922	0.604811	0.302405	−	0.953179i	$-0.402210\pi$
$113$	6.42922	0.604811	0.302405	+	0.953179i	$0.402210\pi$
$114$	0	0
$115$	−10.4328	−0.972866
$116$	−5.55201	−0.515491
$117$	0	0
$118$	−4.70299	−0.432946
$119$	1.77017	0.162271
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.17594	0.106464
$123$	0	0
$124$	−2.60615	−0.234039
$125$	−11.7951	−1.05499
$126$	0	0
$127$	−10.0065	−0.887931	−0.443965	−	0.896044i	$-0.646429\pi$
$127$	−10.0065	−0.887931	−0.443965	+	0.896044i	$0.646429\pi$
$128$	3.54047	0.312936
$129$	0	0
$130$	−15.7944	−1.38526
$131$	−19.8295	−1.73252	−0.866258	−	0.499598i	$-0.833481\pi$
$131$	−19.8295	−1.73252	−0.866258	+	0.499598i	$0.833481\pi$
$132$	0	0
$133$	−5.33965	−0.463006
$134$	−9.54993	−0.824988
$135$	0	0
$136$	6.65616	0.570761
$137$	6.93022	0.592089	0.296044	−	0.955174i	$-0.404332\pi$
$137$	6.93022	0.592089	0.296044	+	0.955174i	$0.404332\pi$
$138$	0	0
$139$	13.4080	1.13725	0.568627	−	0.822595i	$-0.307475\pi$
$139$	13.4080	1.13725	0.568627	+	0.822595i	$0.307475\pi$
$140$	−1.83817	−0.155353
$141$	0	0
$142$	7.93308	0.665729
$143$	−3.69107	−0.308663
$144$	0	0
$145$	−32.7347	−2.71847
$146$	4.88430	0.404227
$147$	0	0
$148$	5.58993	0.459489
$149$	−15.8954	−1.30221	−0.651103	−	0.758990i	$-0.725693\pi$
$149$	−15.8954	−1.30221	−0.651103	+	0.758990i	$0.725693\pi$
$150$	0	0
$151$	19.0406	1.54950	0.774751	−	0.632267i	$-0.217875\pi$
$151$	19.0406	1.54950	0.774751	+	0.632267i	$0.217875\pi$
$152$	−20.0780	−1.62854
$153$	0	0
$154$	0.962480	0.0775589
$155$	−15.3659	−1.23422
$156$	0	0
$157$	−5.61534	−0.448153	−0.224076	−	0.974572i	$-0.571937\pi$
$157$	−5.61534	−0.448153	−0.224076	+	0.974572i	$0.571937\pi$
$158$	−18.5818	−1.47829
$159$	0	0
$160$	−12.1937	−0.963998
$161$	−2.34662	−0.184940
$162$	0	0
$163$	4.53431	0.355154	0.177577	−	0.984107i	$-0.443174\pi$
$163$	4.53431	0.355154	0.177577	+	0.984107i	$0.443174\pi$
$164$	3.47351	0.271236
$165$	0	0
$166$	−1.79338	−0.139193
$167$	4.56122	0.352958	0.176479	−	0.984304i	$-0.443529\pi$
$167$	4.56122	0.352958	0.176479	+	0.984304i	$0.443529\pi$
$168$	0	0
$169$	0.623994	0.0479996
$170$	9.25461	0.709796
$171$	0	0
$172$	−2.94127	−0.224270
$173$	−13.6353	−1.03667	−0.518336	−	0.855177i	$-0.673448\pi$
$173$	−13.6353	−1.03667	−0.518336	+	0.855177i	$0.673448\pi$
$174$	0	0
$175$	−6.74544	−0.509907
$176$	2.38474	0.179757
$177$	0	0
$178$	2.73039	0.204651
$179$	−11.6469	−0.870532	−0.435266	−	0.900302i	$-0.643346\pi$
$179$	−11.6469	−0.870532	−0.435266	+	0.900302i	$0.643346\pi$
$180$	0	0
$181$	−8.32364	−0.618691	−0.309346	−	0.950950i	$-0.600110\pi$
$181$	−8.32364	−0.618691	−0.309346	+	0.950950i	$0.600110\pi$
$182$	−3.55258	−0.263335
$183$	0	0
$184$	−8.82371	−0.650492
$185$	32.9583	2.42314
$186$	0	0
$187$	2.16275	0.158156
$188$	4.04064	0.294694
$189$	0	0
$190$	−27.9161	−2.02525
$191$	16.6026	1.20132	0.600662	−	0.799503i	$-0.294904\pi$
$191$	16.6026	1.20132	0.600662	+	0.799503i	$0.294904\pi$
$192$	0	0
$193$	13.8476	0.996774	0.498387	−	0.866955i	$-0.333926\pi$
$193$	13.8476	0.996774	0.498387	+	0.866955i	$0.333926\pi$
$194$	4.01946	0.288581
$195$	0	0
$196$	3.90678	0.279056
$197$	5.07299	0.361435	0.180718	−	0.983535i	$-0.442158\pi$
$197$	5.07299	0.361435	0.180718	+	0.983535i	$0.442158\pi$
$198$	0	0
$199$	−14.3754	−1.01905	−0.509523	−	0.860457i	$-0.670178\pi$
$199$	−14.3754	−1.01905	−0.509523	+	0.860457i	$0.670178\pi$
$200$	−25.3640	−1.79351
$201$	0	0
$202$	5.48405	0.385857
$203$	−7.36291	−0.516775
$204$	0	0
$205$	20.4798	1.43037
$206$	−1.58481	−0.110419
$207$	0	0
$208$	−8.80225	−0.610326
$209$	−6.52386	−0.451265
$210$	0	0
$211$	28.5168	1.96317	0.981587	−	0.191015i	$-0.0611780\pi$
$211$	28.5168	1.96317	0.981587	+	0.191015i	$0.0611780\pi$
$212$	1.26887	0.0871464
$213$	0	0
$214$	−12.5548	−0.858226
$215$	−17.3418	−1.18270
$216$	0	0
$217$	−3.45620	−0.234622
$218$	−15.0209	−1.01734
$219$	0	0
$220$	−2.24583	−0.151414
$221$	−7.98288	−0.536986
$222$	0	0
$223$	7.22052	0.483522	0.241761	−	0.970336i	$-0.422275\pi$
$223$	7.22052	0.483522	0.241761	+	0.970336i	$0.422275\pi$
$224$	−2.74269	−0.183254
$225$	0	0
$226$	7.56035	0.502907
$227$	−2.02883	−0.134658	−0.0673292	−	0.997731i	$-0.521448\pi$
$227$	−2.02883	−0.134658	−0.0673292	+	0.997731i	$0.521448\pi$
$228$	0	0
$229$	11.9249	0.788018	0.394009	−	0.919107i	$-0.371088\pi$
$229$	11.9249	0.788018	0.394009	+	0.919107i	$0.371088\pi$
$230$	−12.2683	−0.808950
$231$	0	0
$232$	−27.6858	−1.81766
$233$	−6.75719	−0.442678	−0.221339	−	0.975197i	$-0.571043\pi$
$233$	−6.75719	−0.442678	−0.221339	+	0.975197i	$0.571043\pi$
$234$	0	0
$235$	23.8237	1.55408
$236$	2.46831	0.160673
$237$	0	0
$238$	2.08161	0.134931
$239$	−3.50359	−0.226628	−0.113314	−	0.993559i	$-0.536147\pi$
$239$	−3.50359	−0.226628	−0.113314	+	0.993559i	$0.536147\pi$
$240$	0	0
$241$	−0.996561	−0.0641942	−0.0320971	−	0.999485i	$-0.510219\pi$
$241$	−0.996561	−0.0641942	−0.0320971	+	0.999485i	$0.510219\pi$
$242$	1.17594	0.0755920
$243$	0	0
$244$	−0.617176	−0.0395107
$245$	23.0344	1.47161
$246$	0	0
$247$	24.0800	1.53217
$248$	−12.9959	−0.825241
$249$	0	0
$250$	−13.8703	−0.877235
$251$	−21.4443	−1.35355	−0.676775	−	0.736190i	$-0.736623\pi$
$251$	−21.4443	−1.35355	−0.676775	+	0.736190i	$0.736623\pi$
$252$	0	0
$253$	−2.86705	−0.180250
$254$	−11.7670	−0.738325
$255$	0	0
$256$	−13.2566	−0.828539
$257$	1.34904	0.0841507	0.0420753	−	0.999114i	$-0.486603\pi$
$257$	1.34904	0.0841507	0.0420753	+	0.999114i	$0.486603\pi$
$258$	0	0
$259$	7.41320	0.460634
$260$	8.28951	0.514093
$261$	0	0
$262$	−23.3183	−1.44061
$263$	−8.94000	−0.551264	−0.275632	−	0.961263i	$-0.588887\pi$
$263$	−8.94000	−0.551264	−0.275632	+	0.961263i	$0.588887\pi$
$264$	0	0
$265$	7.48127	0.459571
$266$	−6.27908	−0.384995
$267$	0	0
$268$	5.01217	0.306167
$269$	27.1085	1.65284	0.826418	−	0.563057i	$-0.190375\pi$
$269$	27.1085	1.65284	0.826418	+	0.563057i	$0.190375\pi$
$270$	0	0
$271$	−16.8216	−1.02184	−0.510920	−	0.859628i	$-0.670695\pi$
$271$	−16.8216	−1.02184	−0.510920	+	0.859628i	$0.670695\pi$
$272$	5.15761	0.312726
$273$	0	0
$274$	8.14949	0.492329
$275$	−8.24142	−0.496976
$276$	0	0
$277$	6.29869	0.378452	0.189226	−	0.981934i	$-0.439402\pi$
$277$	6.29869	0.378452	0.189226	+	0.981934i	$0.439402\pi$
$278$	15.7670	0.945640
$279$	0	0
$280$	−9.16626	−0.547789
$281$	−5.57640	−0.332660	−0.166330	−	0.986070i	$-0.553192\pi$
$281$	−5.57640	−0.332660	−0.166330	+	0.986070i	$0.553192\pi$
$282$	0	0
$283$	−3.96140	−0.235481	−0.117740	−	0.993044i	$-0.537565\pi$
$283$	−3.96140	−0.235481	−0.117740	+	0.993044i	$0.537565\pi$
$284$	−4.16359	−0.247063
$285$	0	0
$286$	−4.34046	−0.256657
$287$	4.60647	0.271911
$288$	0	0
$289$	−12.3225	−0.724853
$290$	−38.4939	−2.26044
$291$	0	0
$292$	−2.56347	−0.150016
$293$	26.7172	1.56083	0.780417	−	0.625259i	$-0.215007\pi$
$293$	26.7172	1.56083	0.780417	+	0.625259i	$0.215007\pi$
$294$	0	0
$295$	14.5532	0.847320
$296$	27.8749	1.62020
$297$	0	0
$298$	−18.6920	−1.08280
$299$	10.5825	0.612000
$300$	0	0
$301$	−3.90062	−0.224828
$302$	22.3905	1.28843
$303$	0	0
$304$	−15.5577	−0.892296
$305$	−3.63888	−0.208361
$306$	0	0
$307$	−21.3197	−1.21678	−0.608391	−	0.793638i	$-0.708185\pi$
$307$	−21.3197	−1.21678	−0.608391	+	0.793638i	$0.708185\pi$
$308$	−0.505147	−0.0287834
$309$	0	0
$310$	−18.0693	−1.02627
$311$	−18.4438	−1.04585	−0.522926	−	0.852378i	$-0.675159\pi$
$311$	−18.4438	−1.04585	−0.522926	+	0.852378i	$0.675159\pi$
$312$	0	0
$313$	−19.9958	−1.13023	−0.565116	−	0.825011i	$-0.691169\pi$
$313$	−19.9958	−1.13023	−0.565116	+	0.825011i	$0.691169\pi$
$314$	−6.60327	−0.372644
$315$	0	0
$316$	9.75247	0.548619
$317$	29.5416	1.65922	0.829610	−	0.558343i	$-0.188563\pi$
$317$	29.5416	1.65922	0.829610	+	0.558343i	$0.188563\pi$
$318$	0	0
$319$	−8.99583	−0.503670
$320$	−31.6946	−1.77178
$321$	0	0
$322$	−2.75948	−0.153780
$323$	−14.1095	−0.785074
$324$	0	0
$325$	30.4196	1.68738
$326$	5.33205	0.295315
$327$	0	0
$328$	17.3211	0.956399
$329$	5.35858	0.295428
$330$	0	0
$331$	−29.4623	−1.61940	−0.809698	−	0.586847i	$-0.800369\pi$
$331$	−29.4623	−1.61940	−0.809698	+	0.586847i	$0.800369\pi$
$332$	0.941235	0.0516570
$333$	0	0
$334$	5.36370	0.293488
$335$	29.5518	1.61459
$336$	0	0
$337$	11.1591	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$337$	11.1591	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$338$	0.733777	0.0399122
$339$	0	0
$340$	−4.85717	−0.263417
$341$	−4.22270	−0.228672
$342$	0	0
$343$	10.9104	0.589107
$344$	−14.6670	−0.790793
$345$	0	0
$346$	−16.0342	−0.862005
$347$	5.77835	0.310198	0.155099	−	0.987899i	$-0.450430\pi$
$347$	5.77835	0.310198	0.155099	+	0.987899i	$0.450430\pi$
$348$	0	0
$349$	−17.7959	−0.952593	−0.476297	−	0.879285i	$-0.658021\pi$
$349$	−17.7959	−0.952593	−0.476297	+	0.879285i	$0.658021\pi$
$350$	−7.93220	−0.423994
$351$	0	0
$352$	−3.35096	−0.178607
$353$	−33.3495	−1.77502	−0.887508	−	0.460793i	$-0.847565\pi$
$353$	−33.3495	−1.77502	−0.887508	+	0.460793i	$0.847565\pi$
$354$	0	0
$355$	−24.5485	−1.30290
$356$	−1.43301	−0.0759495
$357$	0	0
$358$	−13.6960	−0.723857
$359$	−32.6868	−1.72514	−0.862571	−	0.505936i	$-0.831147\pi$
$359$	−32.6868	−1.72514	−0.862571	+	0.505936i	$0.831147\pi$
$360$	0	0
$361$	23.5607	1.24004
$362$	−9.78806	−0.514449
$363$	0	0
$364$	1.86453	0.0977280
$365$	−15.1142	−0.791115
$366$	0	0
$367$	−2.37352	−0.123896	−0.0619482	−	0.998079i	$-0.519731\pi$
$367$	−2.37352	−0.123896	−0.0619482	+	0.998079i	$0.519731\pi$
$368$	−6.83716	−0.356412
$369$	0	0
$370$	38.7568	2.01487
$371$	1.68274	0.0873635
$372$	0	0
$373$	1.53180	0.0793135	0.0396568	−	0.999213i	$-0.487374\pi$
$373$	1.53180	0.0793135	0.0396568	+	0.999213i	$0.487374\pi$
$374$	2.54326	0.131509
$375$	0	0
$376$	20.1492	1.03912
$377$	33.2042	1.71011
$378$	0	0
$379$	−2.52988	−0.129951	−0.0649756	−	0.997887i	$-0.520697\pi$
$379$	−2.52988	−0.129951	−0.0649756	+	0.997887i	$0.520697\pi$
$380$	14.6515	0.751604
$381$	0	0
$382$	19.5236	0.998915
$383$	11.3646	0.580702	0.290351	−	0.956920i	$-0.406228\pi$
$383$	11.3646	0.580702	0.290351	+	0.956920i	$0.406228\pi$
$384$	0	0
$385$	−2.97835	−0.151791
$386$	16.2839	0.828829
$387$	0	0
$388$	−2.10957	−0.107097
$389$	−31.9767	−1.62128	−0.810642	−	0.585542i	$-0.800882\pi$
$389$	−31.9767	−1.62128	−0.810642	+	0.585542i	$0.800882\pi$
$390$	0	0
$391$	−6.20072	−0.313584
$392$	19.4817	0.983973
$393$	0	0
$394$	5.96551	0.300538
$395$	57.5006	2.89317
$396$	0	0
$397$	−31.3558	−1.57370	−0.786850	−	0.617144i	$-0.788289\pi$
$397$	−31.3558	−1.57370	−0.786850	+	0.617144i	$0.788289\pi$
$398$	−16.9045	−0.847348
$399$	0	0
$400$	−19.6537	−0.982683
$401$	0.0868201	0.00433559	0.00216779	−	0.999998i	$-0.499310\pi$
$401$	0.0868201	0.00433559	0.00216779	+	0.999998i	$0.499310\pi$
$402$	0	0
$403$	15.5863	0.776407
$404$	−2.87824	−0.143198
$405$	0	0
$406$	−8.65831	−0.429705
$407$	9.05727	0.448952
$408$	0	0
$409$	−22.1417	−1.09484	−0.547419	−	0.836859i	$-0.684390\pi$
$409$	−22.1417	−1.09484	−0.547419	+	0.836859i	$0.684390\pi$
$410$	24.0830	1.18937
$411$	0	0
$412$	0.831768	0.0409783
$413$	3.27340	0.161074
$414$	0	0
$415$	5.54953	0.272416
$416$	12.3686	0.606421
$417$	0	0
$418$	−7.67163	−0.375232
$419$	−16.5487	−0.808457	−0.404228	−	0.914658i	$-0.632460\pi$
$419$	−16.5487	−0.808457	−0.404228	+	0.914658i	$0.632460\pi$
$420$	0	0
$421$	−33.8545	−1.64997	−0.824985	−	0.565155i	$-0.808816\pi$
$421$	−33.8545	−1.64997	−0.824985	+	0.565155i	$0.808816\pi$
$422$	33.5339	1.63240
$423$	0	0
$424$	6.32739	0.307285
$425$	−17.8242	−0.864599
$426$	0	0
$427$	−0.818481	−0.0396091
$428$	6.58923	0.318502
$429$	0	0
$430$	−20.3928	−0.983427
$431$	−12.8579	−0.619342	−0.309671	−	0.950844i	$-0.600219\pi$
$431$	−12.8579	−0.619342	−0.309671	+	0.950844i	$0.600219\pi$
$432$	0	0
$433$	27.1158	1.30310	0.651550	−	0.758606i	$-0.274119\pi$
$433$	27.1158	1.30310	0.651550	+	0.758606i	$0.274119\pi$
$434$	−4.06426	−0.195091
$435$	0	0
$436$	7.88355	0.377554
$437$	18.7042	0.894743
$438$	0	0
$439$	8.63388	0.412072	0.206036	−	0.978544i	$-0.433944\pi$
$439$	8.63388	0.412072	0.206036	+	0.978544i	$0.433944\pi$
$440$	−11.1991	−0.533897
$441$	0	0
$442$	−9.38735	−0.446511
$443$	35.1349	1.66931	0.834655	−	0.550773i	$-0.185667\pi$
$443$	35.1349	1.66931	0.834655	+	0.550773i	$0.185667\pi$
$444$	0	0
$445$	−8.44906	−0.400524
$446$	8.49086	0.402054
$447$	0	0
$448$	−7.12896	−0.336812
$449$	17.2210	0.812710	0.406355	−	0.913715i	$-0.366800\pi$
$449$	17.2210	0.812710	0.406355	+	0.913715i	$0.366800\pi$
$450$	0	0
$451$	5.62807	0.265015
$452$	−3.96796	−0.186637
$453$	0	0
$454$	−2.38578	−0.111970
$455$	10.9933	0.515374
$456$	0	0
$457$	−8.99466	−0.420752	−0.210376	−	0.977621i	$-0.567469\pi$
$457$	−8.99466	−0.420752	−0.210376	+	0.977621i	$0.567469\pi$
$458$	14.0229	0.655246
$459$	0	0
$460$	6.43889	0.300215
$461$	−6.06700	−0.282568	−0.141284	−	0.989969i	$-0.545123\pi$
$461$	−6.06700	−0.282568	−0.141284	+	0.989969i	$0.545123\pi$
$462$	0	0
$463$	−15.5851	−0.724302	−0.362151	−	0.932119i	$-0.617957\pi$
$463$	−15.5851	−0.724302	−0.362151	+	0.932119i	$0.617957\pi$
$464$	−21.4527	−0.995918
$465$	0	0
$466$	−7.94601	−0.368092
$467$	10.1017	0.467449	0.233725	−	0.972303i	$-0.424909\pi$
$467$	10.1017	0.467449	0.233725	+	0.972303i	$0.424909\pi$
$468$	0	0
$469$	6.64699	0.306930
$470$	28.0151	1.29224
$471$	0	0
$472$	12.3086	0.566548
$473$	−4.76569	−0.219127
$474$	0	0
$475$	53.7658	2.46695
$476$	−1.09251	−0.0500750
$477$	0	0
$478$	−4.11999	−0.188444
$479$	2.37277	0.108414	0.0542072	−	0.998530i	$-0.482737\pi$
$479$	2.37277	0.108414	0.0542072	+	0.998530i	$0.482737\pi$
$480$	0	0
$481$	−33.4310	−1.52432
$482$	−1.17189	−0.0533782
$483$	0	0
$484$	−0.617176	−0.0280535
$485$	−12.4380	−0.564782
$486$	0	0
$487$	24.2704	1.09980	0.549899	−	0.835231i	$-0.314666\pi$
$487$	24.2704	1.09980	0.549899	+	0.835231i	$0.314666\pi$
$488$	−3.07763	−0.139318
$489$	0	0
$490$	27.0870	1.22366
$491$	0.0568414	0.00256522	0.00128261	−	0.999999i	$-0.499592\pi$
$491$	0.0568414	0.00256522	0.00128261	+	0.999999i	$0.499592\pi$
$492$	0	0
$493$	−19.4558	−0.876244
$494$	28.3165	1.27402
$495$	0	0
$496$	−10.0700	−0.452158
$497$	−5.52162	−0.247679
$498$	0	0
$499$	−13.6648	−0.611722	−0.305861	−	0.952076i	$-0.598944\pi$
$499$	−13.6648	−0.611722	−0.305861	+	0.952076i	$0.598944\pi$
$500$	7.27967	0.325557
$501$	0	0
$502$	−25.2171	−1.12549
$503$	−2.02054	−0.0900915	−0.0450458	−	0.998985i	$-0.514343\pi$
$503$	−2.02054	−0.0900915	−0.0450458	+	0.998985i	$0.514343\pi$
$504$	0	0
$505$	−16.9701	−0.755161
$506$	−3.37146	−0.149880
$507$	0	0
$508$	6.17576	0.274005
$509$	28.5684	1.26627	0.633137	−	0.774040i	$-0.281767\pi$
$509$	28.5684	1.26627	0.633137	+	0.774040i	$0.281767\pi$
$510$	0	0
$511$	−3.39959	−0.150389
$512$	−22.6699	−1.00188
$513$	0	0
$514$	1.58638	0.0699723
$515$	4.90411	0.216101
$516$	0	0
$517$	6.54699	0.287936
$518$	8.71744	0.383022
$519$	0	0
$520$	41.3367	1.81273
$521$	15.9092	0.696994	0.348497	−	0.937310i	$-0.386692\pi$
$521$	15.9092	0.696994	0.348497	+	0.937310i	$0.386692\pi$
$522$	0	0
$523$	−35.5852	−1.55603	−0.778016	−	0.628245i	$-0.783774\pi$
$523$	−35.5852	−1.55603	−0.778016	+	0.628245i	$0.783774\pi$
$524$	12.2383	0.534634
$525$	0	0
$526$	−10.5129	−0.458382
$527$	−9.13266	−0.397825
$528$	0	0
$529$	−14.7800	−0.642611
$530$	8.79749	0.382139
$531$	0	0
$532$	3.29550	0.142878
$533$	−20.7736	−0.899804
$534$	0	0
$535$	38.8501	1.67964
$536$	24.9938	1.07957
$537$	0	0
$538$	31.8779	1.37435
$539$	6.33009	0.272656
$540$	0	0
$541$	−45.1985	−1.94323	−0.971617	−	0.236561i	$-0.923980\pi$
$541$	−45.1985	−1.94323	−0.971617	+	0.236561i	$0.923980\pi$
$542$	−19.7811	−0.849673
$543$	0	0
$544$	−7.24730	−0.310725
$545$	46.4815	1.99105
$546$	0	0
$547$	−36.8981	−1.57765	−0.788825	−	0.614618i	$-0.789310\pi$
$547$	−36.8981	−1.57765	−0.788825	+	0.614618i	$0.789310\pi$
$548$	−4.27717	−0.182712
$549$	0	0
$550$	−9.69137	−0.413242
$551$	58.6875	2.50017
$552$	0	0
$553$	12.9334	0.549986
$554$	7.40685	0.314687
$555$	0	0
$556$	−8.27512	−0.350943
$557$	−15.9844	−0.677280	−0.338640	−	0.940916i	$-0.609967\pi$
$557$	−15.9844	−0.677280	−0.338640	+	0.940916i	$0.609967\pi$
$558$	0	0
$559$	17.5905	0.743998
$560$	−7.10259	−0.300139
$561$	0	0
$562$	−6.55748	−0.276611
$563$	−10.7124	−0.451472	−0.225736	−	0.974188i	$-0.572479\pi$
$563$	−10.7124	−0.451472	−0.225736	+	0.974188i	$0.572479\pi$
$564$	0	0
$565$	−23.3952	−0.984241
$566$	−4.65835	−0.195805
$567$	0	0
$568$	−20.7623	−0.871166
$569$	23.1495	0.970476	0.485238	−	0.874382i	$-0.338733\pi$
$569$	23.1495	0.970476	0.485238	+	0.874382i	$0.338733\pi$
$570$	0	0
$571$	−42.0660	−1.76041	−0.880204	−	0.474596i	$-0.842594\pi$
$571$	−42.0660	−1.76041	−0.880204	+	0.474596i	$0.842594\pi$
$572$	2.27804	0.0952496
$573$	0	0
$574$	5.41691	0.226097
$575$	23.6285	0.985378
$576$	0	0
$577$	18.8051	0.782868	0.391434	−	0.920206i	$-0.371979\pi$
$577$	18.8051	0.782868	0.391434	+	0.920206i	$0.371979\pi$
$578$	−14.4905	−0.602723
$579$	0	0
$580$	20.2031	0.838887
$581$	1.24824	0.0517856
$582$	0	0
$583$	2.05593	0.0851479
$584$	−12.7831	−0.528967
$585$	0	0
$586$	31.4177	1.29785
$587$	−10.6019	−0.437588	−0.218794	−	0.975771i	$-0.570212\pi$
$587$	−10.6019	−0.437588	−0.218794	+	0.975771i	$0.570212\pi$
$588$	0	0
$589$	27.5483	1.13511
$590$	17.1136	0.704556
$591$	0	0
$592$	21.5992	0.887723
$593$	3.70606	0.152189	0.0760947	−	0.997101i	$-0.475755\pi$
$593$	3.70606	0.152189	0.0760947	+	0.997101i	$0.475755\pi$
$594$	0	0
$595$	−6.44144	−0.264073
$596$	9.81029	0.401845
$597$	0	0
$598$	12.4443	0.508885
$599$	−9.78840	−0.399943	−0.199972	−	0.979802i	$-0.564085\pi$
$599$	−9.78840	−0.399943	−0.199972	+	0.979802i	$0.564085\pi$
$600$	0	0
$601$	−44.1815	−1.80220	−0.901101	−	0.433609i	$-0.857240\pi$
$601$	−44.1815	−1.80220	−0.901101	+	0.433609i	$0.857240\pi$
$602$	−4.58688	−0.186947
$603$	0	0
$604$	−11.7514	−0.478158
$605$	−3.63888	−0.147941
$606$	0	0
$607$	29.5291	1.19855	0.599275	−	0.800543i	$-0.295456\pi$
$607$	29.5291	1.19855	0.599275	+	0.800543i	$0.295456\pi$
$608$	21.8612	0.886587
$609$	0	0
$610$	−4.27908	−0.173255
$611$	−24.1654	−0.977627
$612$	0	0
$613$	−13.5655	−0.547905	−0.273953	−	0.961743i	$-0.588331\pi$
$613$	−13.5655	−0.547905	−0.273953	+	0.961743i	$0.588331\pi$
$614$	−25.0706	−1.01177
$615$	0	0
$616$	−2.51898	−0.101493
$617$	−34.0910	−1.37245	−0.686227	−	0.727388i	$-0.740734\pi$
$617$	−34.0910	−1.37245	−0.686227	+	0.727388i	$0.740734\pi$
$618$	0	0
$619$	32.8006	1.31837	0.659184	−	0.751982i	$-0.270902\pi$
$619$	32.8006	1.31837	0.659184	+	0.751982i	$0.270902\pi$
$620$	9.48345	0.380865
$621$	0	0
$622$	−21.6887	−0.869638
$623$	−1.90042	−0.0761387
$624$	0	0
$625$	1.71388	0.0685553
$626$	−23.5138	−0.939801
$627$	0	0
$628$	3.46565	0.138295
$629$	19.5886	0.781051
$630$	0	0
$631$	−32.0061	−1.27414	−0.637071	−	0.770805i	$-0.719854\pi$
$631$	−32.0061	−1.27414	−0.637071	+	0.770805i	$0.719854\pi$
$632$	48.6320	1.93448
$633$	0	0
$634$	34.7390	1.37966
$635$	36.4123	1.44498
$636$	0	0
$637$	−23.3648	−0.925747
$638$	−10.5785	−0.418807
$639$	0	0
$640$	−12.8833	−0.509259
$641$	7.84386	0.309814	0.154907	−	0.987929i	$-0.450492\pi$
$641$	7.84386	0.309814	0.154907	+	0.987929i	$0.450492\pi$
$642$	0	0
$643$	35.1957	1.38798	0.693992	−	0.719982i	$-0.255850\pi$
$643$	35.1957	1.38798	0.693992	+	0.719982i	$0.255850\pi$
$644$	1.44828	0.0570702
$645$	0	0
$646$	−16.5919	−0.652798
$647$	9.21144	0.362139	0.181070	−	0.983470i	$-0.442044\pi$
$647$	9.21144	0.362139	0.181070	+	0.983470i	$0.442044\pi$
$648$	0	0
$649$	3.99937	0.156989
$650$	35.7715	1.40308
$651$	0	0
$652$	−2.79847	−0.109596
$653$	−19.5169	−0.763756	−0.381878	−	0.924213i	$-0.624723\pi$
$653$	−19.5169	−0.763756	−0.381878	+	0.924213i	$0.624723\pi$
$654$	0	0
$655$	72.1573	2.81942
$656$	13.4215	0.524021
$657$	0	0
$658$	6.30134	0.245652
$659$	17.5182	0.682414	0.341207	−	0.939988i	$-0.389164\pi$
$659$	17.5182	0.682414	0.341207	+	0.939988i	$0.389164\pi$
$660$	0	0
$661$	39.1424	1.52246	0.761231	−	0.648481i	$-0.224596\pi$
$661$	39.1424	1.52246	0.761231	+	0.648481i	$0.224596\pi$
$662$	−34.6458	−1.34655
$663$	0	0
$664$	4.69359	0.182147
$665$	19.4303	0.753476
$666$	0	0
$667$	25.7915	0.998650
$668$	−2.81508	−0.108919
$669$	0	0
$670$	34.7510	1.34255
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−49.0838	−1.89204	−0.946022	−	0.324103i	$-0.894937\pi$
$673$	−49.0838	−1.89204	−0.946022	+	0.324103i	$0.894937\pi$
$674$	13.1223	0.505453
$675$	0	0
$676$	−0.385114	−0.0148121
$677$	−18.4538	−0.709237	−0.354619	−	0.935011i	$-0.615389\pi$
$677$	−18.4538	−0.709237	−0.354619	+	0.935011i	$0.615389\pi$
$678$	0	0
$679$	−2.79765	−0.107364
$680$	−24.2209	−0.928830
$681$	0	0
$682$	−4.96562	−0.190143
$683$	29.3885	1.12452	0.562261	−	0.826960i	$-0.309932\pi$
$683$	29.3885	1.12452	0.562261	+	0.826960i	$0.309932\pi$
$684$	0	0
$685$	−25.2182	−0.963539
$686$	12.8299	0.489850
$687$	0	0
$688$	−11.3649	−0.433284
$689$	−7.58858	−0.289102
$690$	0	0
$691$	−30.9601	−1.17778	−0.588890	−	0.808213i	$-0.700435\pi$
$691$	−30.9601	−1.17778	−0.588890	+	0.808213i	$0.700435\pi$
$692$	8.41537	0.319905
$693$	0	0
$694$	6.79497	0.257933
$695$	−48.7902	−1.85072
$696$	0	0
$697$	12.1721	0.461052
$698$	−20.9268	−0.792093
$699$	0	0
$700$	4.16313	0.157351
$701$	−10.2866	−0.388518	−0.194259	−	0.980950i	$-0.562230\pi$
$701$	−10.2866	−0.388518	−0.194259	+	0.980950i	$0.562230\pi$
$702$	0	0
$703$	−59.0883	−2.22856
$704$	−8.70999	−0.328270
$705$	0	0
$706$	−39.2169	−1.47595
$707$	−3.81704	−0.143555
$708$	0	0
$709$	5.76859	0.216644	0.108322	−	0.994116i	$-0.465452\pi$
$709$	5.76859	0.216644	0.108322	+	0.994116i	$0.465452\pi$
$710$	−28.8675	−1.08338
$711$	0	0
$712$	−7.14591	−0.267804
$713$	12.1067	0.453398
$714$	0	0
$715$	13.4313	0.502304
$716$	7.18820	0.268636
$717$	0	0
$718$	−38.4375	−1.43448
$719$	−27.7456	−1.03474	−0.517368	−	0.855763i	$-0.673088\pi$
$719$	−27.7456	−1.03474	−0.517368	+	0.855763i	$0.673088\pi$
$720$	0	0
$721$	1.10307	0.0410803
$722$	27.7059	1.03111
$723$	0	0
$724$	5.13715	0.190921
$725$	74.1384	2.75343
$726$	0	0
$727$	8.82510	0.327305	0.163652	−	0.986518i	$-0.447672\pi$
$727$	8.82510	0.327305	0.163652	+	0.986518i	$0.447672\pi$
$728$	9.29773	0.344597
$729$	0	0
$730$	−17.7733	−0.657821
$731$	−10.3070	−0.381219
$732$	0	0
$733$	1.10562	0.0408370	0.0204185	−	0.999792i	$-0.493500\pi$
$733$	1.10562	0.0408370	0.0204185	+	0.999792i	$0.493500\pi$
$734$	−2.79110	−0.103021
$735$	0	0
$736$	9.60735	0.354132
$737$	8.12113	0.299146
$738$	0	0
$739$	−21.6890	−0.797844	−0.398922	−	0.916985i	$-0.630616\pi$
$739$	−21.6890	−0.797844	−0.398922	+	0.916985i	$0.630616\pi$
$740$	−20.3411	−0.747752
$741$	0	0
$742$	1.97879	0.0726437
$743$	−3.74975	−0.137565	−0.0687825	−	0.997632i	$-0.521911\pi$
$743$	−3.74975	−0.137565	−0.0687825	+	0.997632i	$0.521911\pi$
$744$	0	0
$745$	57.8415	2.11915
$746$	1.80130	0.0659501
$747$	0	0
$748$	−1.33480	−0.0488051
$749$	8.73844	0.319296
$750$	0	0
$751$	−13.4176	−0.489617	−0.244808	−	0.969571i	$-0.578725\pi$
$751$	−13.4176	−0.489617	−0.244808	+	0.969571i	$0.578725\pi$
$752$	15.6129	0.569343
$753$	0	0
$754$	39.0460	1.42197
$755$	−69.2864	−2.52159
$756$	0	0
$757$	−19.8967	−0.723159	−0.361580	−	0.932341i	$-0.617762\pi$
$757$	−19.8967	−0.723159	−0.361580	+	0.932341i	$0.617762\pi$
$758$	−2.97497	−0.108056
$759$	0	0
$760$	73.0614	2.65022
$761$	16.2938	0.590648	0.295324	−	0.955397i	$-0.404572\pi$
$761$	16.2938	0.590648	0.295324	+	0.955397i	$0.404572\pi$
$762$	0	0
$763$	10.4549	0.378494
$764$	−10.2467	−0.370714
$765$	0	0
$766$	13.3640	0.482861
$767$	−14.7619	−0.533023
$768$	0	0
$769$	−46.7621	−1.68628	−0.843142	−	0.537691i	$-0.819297\pi$
$769$	−46.7621	−1.68628	−0.843142	+	0.537691i	$0.819297\pi$
$770$	−3.50235	−0.126216
$771$	0	0
$772$	−8.54642	−0.307592
$773$	12.2550	0.440782	0.220391	−	0.975412i	$-0.429267\pi$
$773$	12.2550	0.440782	0.220391	+	0.975412i	$0.429267\pi$
$774$	0	0
$775$	34.8010	1.25009
$776$	−10.5196	−0.377633
$777$	0	0
$778$	−37.6025	−1.34812
$779$	−36.7167	−1.31551
$780$	0	0
$781$	−6.74619	−0.241398
$782$	−7.29164	−0.260749
$783$	0	0
$784$	15.0956	0.539129
$785$	20.4335	0.729304
$786$	0	0
$787$	8.61710	0.307166	0.153583	−	0.988136i	$-0.450919\pi$
$787$	8.61710	0.307166	0.153583	+	0.988136i	$0.450919\pi$
$788$	−3.13093	−0.111535
$789$	0	0
$790$	67.6170	2.40571
$791$	−5.26220	−0.187102
$792$	0	0
$793$	3.69107	0.131074
$794$	−36.8723	−1.30855
$795$	0	0
$796$	8.87215	0.314465
$797$	−24.4325	−0.865443	−0.432721	−	0.901528i	$-0.642447\pi$
$797$	−24.4325	−0.865443	−0.432721	+	0.901528i	$0.642447\pi$
$798$	0	0
$799$	14.1595	0.500928
$800$	27.6166	0.976396
$801$	0	0
$802$	0.102095	0.00360509
$803$	−4.15354	−0.146575
$804$	0	0
$805$	8.53907	0.300963
$806$	18.3284	0.645592
$807$	0	0
$808$	−14.3527	−0.504927
$809$	18.9531	0.666357	0.333178	−	0.942864i	$-0.391879\pi$
$809$	18.9531	0.666357	0.333178	+	0.942864i	$0.391879\pi$
$810$	0	0
$811$	−49.0218	−1.72139	−0.860695	−	0.509121i	$-0.829971\pi$
$811$	−49.0218	−1.72139	−0.860695	+	0.509121i	$0.829971\pi$
$812$	4.54421	0.159471
$813$	0	0
$814$	10.6508	0.373309
$815$	−16.4998	−0.577962
$816$	0	0
$817$	31.0907	1.08772
$818$	−26.0372	−0.910371
$819$	0	0
$820$	−12.6397	−0.441397
$821$	33.5837	1.17208	0.586039	−	0.810283i	$-0.300687\pi$
$821$	33.5837	1.17208	0.586039	+	0.810283i	$0.300687\pi$
$822$	0	0
$823$	51.4914	1.79488	0.897438	−	0.441141i	$-0.145426\pi$
$823$	51.4914	1.79488	0.897438	+	0.441141i	$0.145426\pi$
$824$	4.14772	0.144493
$825$	0	0
$826$	3.84931	0.133935
$827$	45.1149	1.56880	0.784400	−	0.620256i	$-0.212971\pi$
$827$	45.1149	1.56880	0.784400	+	0.620256i	$0.212971\pi$
$828$	0	0
$829$	−34.8361	−1.20991	−0.604954	−	0.796261i	$-0.706808\pi$
$829$	−34.8361	−1.20991	−0.604954	+	0.796261i	$0.706808\pi$
$830$	6.52589	0.226517
$831$	0	0
$832$	32.1492	1.11457
$833$	13.6904	0.474345
$834$	0	0
$835$	−16.5977	−0.574387
$836$	4.02637	0.139255
$837$	0	0
$838$	−19.4602	−0.672241
$839$	−24.6656	−0.851552	−0.425776	−	0.904829i	$-0.639999\pi$
$839$	−24.6656	−0.851552	−0.425776	+	0.904829i	$0.639999\pi$
$840$	0	0
$841$	51.9250	1.79052
$842$	−39.8108	−1.37197
$843$	0	0
$844$	−17.5999	−0.605812
$845$	−2.27064	−0.0781123
$846$	0	0
$847$	−0.818481	−0.0281233
$848$	4.90286	0.168365
$849$	0	0
$850$	−20.9601	−0.718924
$851$	−25.9676	−0.890158
$852$	0	0
$853$	18.8895	0.646764	0.323382	−	0.946269i	$-0.395180\pi$
$853$	18.8895	0.646764	0.323382	+	0.946269i	$0.395180\pi$
$854$	−0.962480	−0.0329354
$855$	0	0
$856$	32.8580	1.12306
$857$	−33.5204	−1.14503	−0.572517	−	0.819893i	$-0.694033\pi$
$857$	−33.5204	−1.14503	−0.572517	+	0.819893i	$0.694033\pi$
$858$	0	0
$859$	57.9945	1.97875	0.989374	−	0.145391i	$-0.0464441\pi$
$859$	57.9945	1.97875	0.989374	+	0.145391i	$0.0464441\pi$
$860$	10.7029	0.364966
$861$	0	0
$862$	−15.1200	−0.514990
$863$	−12.9830	−0.441947	−0.220973	−	0.975280i	$-0.570923\pi$
$863$	−12.9830	−0.441947	−0.220973	+	0.975280i	$0.570923\pi$
$864$	0	0
$865$	49.6171	1.68703
$866$	31.8864	1.08354
$867$	0	0
$868$	2.13308	0.0724015
$869$	15.8018	0.536038
$870$	0	0
$871$	−29.9757	−1.01569
$872$	39.3124	1.33128
$873$	0	0
$874$	21.9949	0.743990
$875$	9.65408	0.326367
$876$	0	0
$877$	17.8721	0.603496	0.301748	−	0.953388i	$-0.402430\pi$
$877$	17.8721	0.603496	0.301748	+	0.953388i	$0.402430\pi$
$878$	10.1529	0.342643
$879$	0	0
$880$	−8.67778	−0.292528
$881$	−57.1705	−1.92612	−0.963061	−	0.269283i	$-0.913213\pi$
$881$	−57.1705	−1.92612	−0.963061	+	0.269283i	$0.913213\pi$
$882$	0	0
$883$	43.3336	1.45829	0.729145	−	0.684359i	$-0.239918\pi$
$883$	43.3336	1.45829	0.729145	+	0.684359i	$0.239918\pi$
$884$	4.92684	0.165708
$885$	0	0
$886$	41.3164	1.38805
$887$	42.2026	1.41702	0.708512	−	0.705698i	$-0.249367\pi$
$887$	42.2026	1.41702	0.708512	+	0.705698i	$0.249367\pi$
$888$	0	0
$889$	8.19010	0.274687
$890$	−9.93554	−0.333040
$891$	0	0
$892$	−4.45633	−0.149209
$893$	−42.7116	−1.42929
$894$	0	0
$895$	42.3817	1.41666
$896$	−2.89781	−0.0968090
$897$	0	0
$898$	20.2508	0.675778
$899$	37.9867	1.26693
$900$	0	0
$901$	4.44647	0.148133
$902$	6.61825	0.220363
$903$	0	0
$904$	−19.7868	−0.658098
$905$	30.2887	1.00683
$906$	0	0
$907$	−2.24837	−0.0746560	−0.0373280	−	0.999303i	$-0.511885\pi$
$907$	−2.24837	−0.0746560	−0.0373280	+	0.999303i	$0.511885\pi$
$908$	1.25215	0.0415540
$909$	0	0
$910$	12.9274	0.428539
$911$	7.38933	0.244819	0.122410	−	0.992480i	$-0.460938\pi$
$911$	7.38933	0.244819	0.122410	+	0.992480i	$0.460938\pi$
$912$	0	0
$913$	1.52507	0.0504724
$914$	−10.5771	−0.349860
$915$	0	0
$916$	−7.35975	−0.243173
$917$	16.2301	0.535965
$918$	0	0
$919$	2.99412	0.0987669	0.0493834	−	0.998780i	$-0.484274\pi$
$919$	2.99412	0.0987669	0.0493834	+	0.998780i	$0.484274\pi$
$920$	32.1084	1.05858
$921$	0	0
$922$	−7.13440	−0.234959
$923$	24.9006	0.819615
$924$	0	0
$925$	−74.6447	−2.45430
$926$	−18.3271	−0.602265
$927$	0	0
$928$	30.1447	0.989547
$929$	34.8125	1.14216	0.571081	−	0.820893i	$-0.306524\pi$
$929$	34.8125	1.14216	0.571081	+	0.820893i	$0.306524\pi$
$930$	0	0
$931$	−41.2966	−1.35344
$932$	4.17037	0.136605
$933$	0	0
$934$	11.8789	0.388690
$935$	−7.87000	−0.257376
$936$	0	0
$937$	−0.426908	−0.0139465	−0.00697324	−	0.999976i	$-0.502220\pi$
$937$	−0.426908	−0.0139465	−0.00697324	+	0.999976i	$0.502220\pi$
$938$	7.81643	0.255216
$939$	0	0
$940$	−14.7034	−0.479572
$941$	5.76575	0.187958	0.0939790	−	0.995574i	$-0.470041\pi$
$941$	5.76575	0.187958	0.0939790	+	0.995574i	$0.470041\pi$
$942$	0	0
$943$	−16.1359	−0.525458
$944$	9.53745	0.310418
$945$	0	0
$946$	−5.60414	−0.182206
$947$	−45.3443	−1.47349	−0.736747	−	0.676169i	$-0.763639\pi$
$947$	−45.3443	−1.47349	−0.736747	+	0.676169i	$0.763639\pi$
$948$	0	0
$949$	15.3310	0.497666
$950$	63.2251	2.05129
$951$	0	0
$952$	−5.44794	−0.176569
$953$	−51.3844	−1.66450	−0.832252	−	0.554397i	$-0.812949\pi$
$953$	−51.3844	−1.66450	−0.832252	+	0.554397i	$0.812949\pi$
$954$	0	0
$955$	−60.4149	−1.95498
$956$	2.16233	0.0699348
$957$	0	0
$958$	2.79022	0.0901479
$959$	−5.67225	−0.183167
$960$	0	0
$961$	−13.1688	−0.424801
$962$	−39.3127	−1.26749
$963$	0	0
$964$	0.615054	0.0198096
$965$	−50.3898	−1.62210
$966$	0	0
$967$	−56.0596	−1.80276	−0.901378	−	0.433033i	$-0.857443\pi$
$967$	−56.0596	−1.80276	−0.901378	+	0.433033i	$0.857443\pi$
$968$	−3.07763	−0.0989188
$969$	0	0
$970$	−14.6263	−0.469623
$971$	20.4358	0.655816	0.327908	−	0.944710i	$-0.393656\pi$
$971$	20.4358	0.655816	0.327908	+	0.944710i	$0.393656\pi$
$972$	0	0
$973$	−10.9742	−0.351817
$974$	28.5404	0.914494
$975$	0	0
$976$	−2.38474	−0.0763337
$977$	−18.3998	−0.588660	−0.294330	−	0.955704i	$-0.595097\pi$
$977$	−18.3998	−0.588660	−0.294330	+	0.955704i	$0.595097\pi$
$978$	0	0
$979$	−2.32189	−0.0742078
$980$	−14.2163	−0.454123
$981$	0	0
$982$	0.0668418	0.00213301
$983$	19.5798	0.624498	0.312249	−	0.950000i	$-0.398918\pi$
$983$	19.5798	0.624498	0.312249	+	0.950000i	$0.398918\pi$
$984$	0	0
$985$	−18.4600	−0.588184
$986$	−22.8787	−0.728607
$987$	0	0
$988$	−14.8616	−0.472811
$989$	13.6635	0.434472
$990$	0	0
$991$	40.0516	1.27228	0.636141	−	0.771573i	$-0.280530\pi$
$991$	40.0516	1.27228	0.636141	+	0.771573i	$0.280530\pi$
$992$	14.1501	0.449266
$993$	0	0
$994$	−6.49307	−0.205948
$995$	52.3103	1.65835
$996$	0	0
$997$	28.4921	0.902353	0.451177	−	0.892435i	$-0.351004\pi$
$997$	28.4921	0.902353	0.451177	+	0.892435i	$0.351004\pi$
$998$	−16.0690	−0.508654
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.10		13
3.2	odd	2		2013.2.a.g.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.4	✓	13	3.2	odd	2
6039.2.a.h.1.10		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17594 q^{2} -0.617176 q^{4} -3.63888 q^{5} -0.818481 q^{7} -3.07763 q^{8} +O(q^{10})\)	\(q+1.17594 q^{2} -0.617176 q^{4} -3.63888 q^{5} -0.818481 q^{7} -3.07763 q^{8} -4.27908 q^{10} -1.00000 q^{11} +3.69107 q^{13} -0.962480 q^{14} -2.38474 q^{16} -2.16275 q^{17} +6.52386 q^{19} +2.24583 q^{20} -1.17594 q^{22} +2.86705 q^{23} +8.24142 q^{25} +4.34046 q^{26} +0.505147 q^{28} +8.99583 q^{29} +4.22270 q^{31} +3.35096 q^{32} -2.54326 q^{34} +2.97835 q^{35} -9.05727 q^{37} +7.67163 q^{38} +11.1991 q^{40} -5.62807 q^{41} +4.76569 q^{43} +0.617176 q^{44} +3.37146 q^{46} -6.54699 q^{47} -6.33009 q^{49} +9.69137 q^{50} -2.27804 q^{52} -2.05593 q^{53} +3.63888 q^{55} +2.51898 q^{56} +10.5785 q^{58} -3.99937 q^{59} +1.00000 q^{61} +4.96562 q^{62} +8.70999 q^{64} -13.4313 q^{65} -8.12113 q^{67} +1.33480 q^{68} +3.50235 q^{70} +6.74619 q^{71} +4.15354 q^{73} -10.6508 q^{74} -4.02637 q^{76} +0.818481 q^{77} -15.8018 q^{79} +8.67778 q^{80} -6.61825 q^{82} -1.52507 q^{83} +7.87000 q^{85} +5.60414 q^{86} +3.07763 q^{88} +2.32189 q^{89} -3.02107 q^{91} -1.76947 q^{92} -7.69883 q^{94} -23.7395 q^{95} +3.41810 q^{97} -7.44378 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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